(a) Let A be the subset of R defined by A= {1-:n€N}. Find all cluster points of A and justify your answer. (b) Let ScR and let e eR be a cluster point of S. Prove that for each e> 0 the set W:=sn(e-t.c+e) has infinitely many elements. Let (z.) be a sequence of real numbers and let B be the subset of R defined by B:= {!:n€N}. Let LE R and let f : B→R be defined by f () = a,-. Prove that lim a, = L if and only if limf(r) = L.

(a) Let A be the subset of R defined by A= {1-:n€N}. Find all cluster points of A and justify your answer. (b) Let ScR and let e eR be a cluster point of S. Prove that for each e> 0 the set W:=sn(e-t.c+e) has infinitely many elements. Let (z.) be a sequence of real numbers and let B be the subset of R defined by B:= {!:n€N}. Let LE R and let f : B→R be defined by f () = a,-. Prove that lim a, = L if and only if limf(r) = L.

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Description: (a) Let A be the subset of R defined by A= {1-:n€N}. Find all cluster points ofA and justify your answer.(b) Let ScR and let e eR be a cluster point of S. Prove that for each e> 0 the setW:=sn(e-t.c+e) has infinitely many elements.Let (z.) be a sequ

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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Question

(a) Let A be the subset of R defined by A= {1-:n€N}. Find all cluster points of
A and justify your answer.
(b) Let ScR and let e eR be a cluster point of S. Prove that for each e> 0 the set
W:=sn(e-t.c+e) has infinitely many elements.
Let (z.) be a sequence of real numbers and let B be the subset of R defined by B:=
{!:n€N}. Let LE R and let f : B→R be defined by f () = a,-. Prove that lim a, = L if
and only if limf(r) = L.

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